Compound Poisson distribution

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In probability theory, a compound Poisson distribution is the probability distribution of a "Poisson-distributed number" of independent identically-distributed random variables. More precisely, suppose

[N\sim\operatorname(\lambda),]

i.e., N is a random variable whose distribution is a Poisson distribution with expected value λ, and

[X_1, X_2, X_3, \dots]

are identically distributed random variables that are mutually independent and also independent of N. Then the probability distribution of the sum

[Y=\sum_^N X_n]

is a compound Poisson distribution. (When N = 0, then the value of Y is 0.)

Via the law of total cumulance it can be shown that the moments of X₁ are the cumulants of Y.

It can be shown that every infinitely divisible probability distribution is a limit of compound Poisson distributions.

Compound Poisson processes

A compound Poisson process with rate [\lambda>0] and jump size distribution G is a continuous-time stochastic process [\] given by

[Y(t) = \sum_^ D_i]

where, [ \] is a Poisson process with rate [\lambda], and [ \] are independent and identically distributed random variables, with distribution function G, which are also independent of [ \.\,]

Probability distributions [ view] • [ talk] • [ edit]
Univariate Multivariate
Discrete: Bernoulli • binomial • Boltzmann • compound Poisson • degenerate • degree • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot Ewens • multinomial
Continuous: Beta • Beta prime • Cauchy • chi-square • exponential • exponential power • F • fading • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square • inverse gaussian • inverse gamma • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • normal (Gaussian) • Pareto • Pearson • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • Student's t • triangular • type-1 Gumbel • type-2 Gumbel • uniform • Voigt • von Mises • Weibull • Wigner semicircle Dirichlet • Kent • matrix normal • multivariate normal • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous: Cantor • conditional • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • phase-type • posterior • prior • quasi • sampling

	Probability distributions [ view] • [ talk] • [ edit]
Univariate	Multivariate
Discrete:	Bernoulli • binomial • Boltzmann • compound Poisson • degenerate • degree • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot	Ewens • multinomial
Continuous:	Beta • Beta prime • Cauchy • chi-square • exponential • exponential power • F • fading • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square • inverse gaussian • inverse gamma • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • normal (Gaussian) • Pareto • Pearson • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • Student's t • triangular • type-1 Gumbel • type-2 Gumbel • uniform • Voigt • von Mises • Weibull • Wigner semicircle	Dirichlet • Kent • matrix normal • multivariate normal • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous:	Cantor • conditional • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • phase-type • posterior • prior • quasi • sampling

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